CR- Submanifolds of a Nearly Trans-Hyperbolic Sasakian Manifold with a Quarter Symmetric Semi Metric Connection

Jurnal Matematika Vol. 6 No. 2, Desember 2016. ISSN: 1693-1394

CR- Submanifolds of a Nearly Trans-Hyperbolic Sasakian Manifold with a Quarter Symmetric Semi Metric Connection

Shamsur Rahman

Department of Mathematics,

Maulana Azad National Urdu University Polytechnic, Darbhanga (Campus) Bihar 846001 India e-mail: shamsur@rediffmail.com

Abstract: The object of the present paper is to initiate the study contact CR-submanifolds of a nearly trans-hyperbolic Sasakian manifold with a quarter symmetric semi metric connection. For this, some properties of CR- submanifolds of a nearly trans-hyperbolic Sasakian manifold with a quarter symmetric semi metric connection are investigated which conclude that CR- submanifolds of a nearly trans-hyperbolic Sasakian manifold with a quarter symmetric semi metric connection exists with respect to the ^-horizontal and ^-vertical.

Keyword: CR-submanifolds, nearly trans-hyperbolic Sasakian manifold, quarter symmetric semi metric connection, Gauss and Weingarten equations parallel distributions.

• 1.    Introduction

Bejancu [1] defined the notion of CR-submanifolds of a Kaehler manifold in [2]. After that a number of authors have studied these submanifolds ([10], [13], [14], [18]). Upadhyay and Dube [15] have defined almost contact hyperbolic (f ,g,η, ξ) -structure, Dube and Mishra [5] have considered Hypersurfaces immersed in an almost hyperbolic Hermitian manifold also Dube and Niwas [6] worked with almost r-contact hyperbolic structure in a product manifold. Gherghe studied on harmonicity on nearly trans-Sasaki manifolds [7]. Bhatt and Dube [3] studied on CR-submanifolds of trans- hyperbolic Sasakian manifold. Joshi and Dube [9] studied on Semi-invariant submanifold of an almost r-contact hyperbolic metric manifold. Gill and Dube have also worked on CR submanifolds of trans-hyperbolic Sasakian manifolds [8].

Let V be a linear connection in an n-dimensional differentiable manifold M. The torsion tensor T and the curvature tensor R of V are given respectively by [4]

T (X,Y}=V_xY -V_γX-[X,Y∖

R (X , Y)Z = V_xV_γZ - V_xV_γZ - V_ftY]Z

The connection V is symmetric if the torsion tensor T vanishes, otherwise it is non-symmetric. The connection V is metric if there is a Riemannian metric g in M such that Vg = 0, otherwise it is non-metric. It is well known that a linear connection is symmetric and metric if and only if it is the Levi-Civita connection. In [17], S. Golab introduced the idea of a quarter-symmetric connection. A linear connection is said to be a quarter-symmetric connection if its torsion tensor is of the form

T (X,Y) = η(Y )φX- η(X)φ Y,

where η is a 1-form. In [11], M. Ahmad, J. B. Jun and A. Haseeb studied some properties of hypersurfaces of an almost r-paracontact Riemannian manifold with quarter symmetric semi metric connection. In [12], M. Ahmad, C. Ozgur and A. Haseeb studied properties of hypersurfaces of an almost r-paracontact Riemannian manifold with quarter symmetric non-metric connection.

In this paper, CR-submanifolds of a nearly trans-hyperbolic Sasakian manifold with a quarter symmetric semi metric connection are investigated. Parallel distribution relating to ξ-vertical CR-submanifolds of a nearly trans-hyperbolic sasakian manifold with a quarter symmetric semi metric connection are also discussed.

• 2.    Preliminaries

Let be an dimensional almost hyperbolic contact metric manifold with the almost hyperbolic contact metric structure (φ, ξ, η, g) where a tensor φ of type (1, 1), a vector field ξ, called structure vector field and η, the dual 1 -form of is a 1-form ξ satisfying the following

• (2.1)    φ² X = X- η(X)ξ, g(X,ξ) = η(X)

• (2.2)    φ(ξ) = 0, η o φ = 0, η(ξ) = -1

• (2.3)    g(φX, φ Y) = -g(X,Y)-η(X)η( Y),

for any X, Y tangents to M [4]. In this case

• (2.4)    gφφX,Y) = -g(X,φ Y)

An almost hyperbolic contact metric structure (φ, ξ, η, g) on M is called trans-hyperbolic Sasakian [6] if and only if _

• (2.5)   (Vχφ)Y = a{g{X, Y)ξ - η(Y)φX} + β{g(φX, Y)ξ - η( Y)φX}

for all X, Y tangents to M and a, β are functions on Mo On a trans-hyperbolic Sasakian manifold M, we have _

• (2.6)    Vχξ = -a(φX) + β{X - η(X)}}

_ a Riemannian metric g and Riemannian connection V.

Further, an almost contact metric manifold M on (φ, ξ, η, g) is called nearly trans-hyperbolic Sasakian if [5]

_ _

• (2.7)  (‰φ)Y + (VγφX = α{2g(X, Y)ξ - η(Y)φX - η (X)φY} - β{η(X)φY +

η (Y)φX}

On other hand, a quarter symmetric semi metric connection V on M is defined by _ _

• (2.8)    V_xY = V^Y - ηXXφφY + gφχx, Yξξ

Using (2.1), (2.2) and (2.6) in (2.5) and (2.6), we get respectively

• (2.9)    (Vχφ)Y = a{g (X, Yξξ - η(Y)φX}

+β{g(φX, Yξξ - ((Y)XX} - gXX, Y)ξ - η^η(Yξξ

• (2.10)   Vχξ = -aφX + β{X-η(X)ξ}

In particular, an almost contact metric manifold M on (φ, ξ, rg g) is called nearly trans-hyperbolic Sasakian manifold M with a quarter symmetric semi metric connection if _ _ _ _ _

• (2.11)   (Vχφ)Y + (Vγφ)X = a{ggtX, Y)ξ - ηQφXX - ηWφY] - β{η(X)φY +

η(Γ)XX} -2η(X)η(F)ξ - 2g(X, Y)ξ

Now, let M be a submanifold immersed in M. The Riemannian metric induced on M is denoted by the same symbol g. Let TM and T^lM be the Lie algebras of vector fields tangential to M and normal to M respectively and V be the induced Levi-Civita connection on M, then the Gauss and

Weingarten formulas for the quarter symmetric semi metric connection are given by

• (2.12)    VjY = V_x Y + X(X,Y)

• (2.13)    V_xN = -A_nX + V^N - η(XφφN

for any X,Y e TM and V e T¹M. where V^l is the connection on the normal bundle T^^lM, h is the second fundamental form and A_n is the Weingarten map associated with

as

• (2.14)     g(A nX,Y) =g(h(X,Y),V)

For any x e M and X e X_xM, we write

• (2.15)    X = PX+ QX

where       and        .

Similarly for normal to , we have

• (2.16)    NN = NN+ CN

where (respectly ) is the tangential component (respectly normal component) of φ N.

Definition. An m dimensional Riemannian submanifold M of M is called a CR-submanifold of M if there exists a differentiable distribution D : x → D_x on M satisfying the following conditions:

• (i)    D is invariant, that is DD_x ⊂ D_x for each x e M,

• (ii)    The complementary orthogonal distribution D¹-: → Dχ⊂ T_xM of D is antiinvariant, that is, ΦDχ ⊂Тx M for each x e M. If dimDχ =0 (respectly dimD_x=0 ), then the CR-submanifold is called an invariant (respectly, anti-invariant) submanifold. The distribution D (respectly, D^l ) is called the horizontal (respectly, vertical) distribution. Also, the pair (D, D¹) is called ξ -horizontal (respectly, vertical) if (      _( respectly, ξχGDx ).

• 3. Some Basic Lemmas

Lemma 1. Let M be a CR-submanifold of a nearly trans-hyperbolic Sasakian manifold _

with a quarter symmetric semi metric connection. Then

• (3.1)    p∇X(φPY)+P∇γ( φPX)-PАφQγX ^-pАφQχY

=2(a-1)g(X,Y)Pξ — aη (Y) φPX - aη (X)φPY - βη (Y) φPX - βη (x)φPY -4η(X)η(Y)Pξ + φP ∇_xY + φP ∇_γX

• (3.2)    Q∇X(φPY)+Q∇Y( φPX)-QАφQγX ^-QАφQχY

=2Bℎ(X,Y)+2(a-1)g(X,Y)Qξ — aη (Y)ΦQX - aη (X)ΦQY

+g(x)QY +g(Y) QX-4g(x)g(Y)Qξ

• (3.3)    ℎ(X, φPY)+ℎ(Y, φPX)+DxΦQY+D ⅛ΦQX

=   ∇γX + ΦQ∇χY+2Cℎ(X,Y)-βg (Y) ΦQX - βg (x)ΦQY

for any x,YeTM.

Proof. Using (2.4) (2.9), and (2.10) in (2.11) we get

(∇xφPY)+ℎ(X, φPY)-АφQγX+DxΦQY ^-Φ (∇χY)-φℎ(X,γ)-g(x) QY +g(x)g(Y)ξ + (v_γφPX)+ℎ(Y, φPX)-АφQχY+D ⅛ΦQX ^-Φ (∇ γX )

-φℎ(Y,X)-g(Y) QX +g(x)g(Y)ξ = a{2 g(X,Y)ξ-g(Y) φX -g(x) φY}

• -β{g(x) φY +g(Y) ψx}-2g(x)g(Y)ξ-2g(x,Y)ξ

Again using (2.15) we get

• (3.4)    P (∇ xΦPY)+P (∇_γφPX)-PАφQγX ^-pАφQχY ^- φP ∇χY ^-g(x)QY

-ΦQ∇χY - φP ∇γX - ΦQ∇γX +Q (∇xφPY)+Q (∇_γφPX)+2g(x)g(Y)Pξ +2g(x)g(Y)Q^{ξ -}QАφQγX ^-QАφQχY+ℎ(X, φPY)+ℎ(Y, φPX)+DxΦQY -g(Y) QX+D ⅛ΦQX -2Bℎ(x,Y)-2Cℎ(X,Y)=2ag (X,Y)Pξ +

2 ag (X,Y)Qξ

• -    aη (Y) φPX - aη (Y)ΦQX - aη (X) φPY - aη (X)ΦQY - βg (x)φPY

• -    βg (x)ΦQY - βg (Y) φPX - βg (Y) ΦQX-2g(x)g(Y)Pξ-2g(x)g(Y)Qξ

• - 2g(x,Y)Pξ-2g(x,Y)Qξ

for any x,YeTM.

Now equating horizontal, vertical, and normal components in (3.4), we get the desired result.

Lemma 2. Let M be a CR-submanifold of a nearly trans-hyperbolic Sasakian manifold M with a quarter symmetric semi metric connection. Then _

• (3.5)    2(X_xφ)Y = X_xφY - X_γφX + h(X,φY) - h(Y,φX) - φ[X,Y]

+a{2g(X, Y)ξ - η(Y)φX - η(X)φY} - β{η(X)φY + η(Y)φX} -2η(X)η(Y)ξ-2g(X,Y)ξ _

• (3.6)    2(V_γφ)X = a{2g(X, Y)ξ - η(Y)φX - η(X)φY} - β{η(X)φY + η(Y)φX}

-2η(X)η(Y)ξ - 2g(X, Y)ξ-XχφY + Y_γφX + h(X, φY) + h(Y, φX) + φ[X,Y]

Proof. From Gauss formula (2.12), we have

• (3.7)   X_xφY - V_γφX = X_xφY + h(X, φY^) - V_γφX - h(Y, φX)

Also we have

• (3.8)   VχφY - ^γφX = (Vχφ)Y - (Xγφ)X + φ[X, Y]

From (3.6) and (3.7), we get _ _

• (3.9)   (Vχφ)Y - (Xγφ)X = X_xφY + h(X, φY) - X_γφX - h(Y, φX) - φ[X, Y]

Also for nearly trans-hyperbolic Sasakian manifold M with a quarter symmetric semi metric connection, we have _ _

• (3.10)  (Vχφ)Y + (Vγφ)X = a{2g(X, Y)ξ - η(Y)φX - η(X)φY} - β{η(X)φY +

η(Y)φX} -2η(X)η(Y)ξ - 2g(X, Y)ξ

Adding (3.9) and (3.10), we get _

2(^χφ)Y = X_xφY - X_γφX + h(X,φY) - h(Y,φX) - φ[X,Y]

+a{2g(X, Y)ξ - η(Y)φX - η(X)φY} - β{η(X)φY + η(Y)φX}

-2η(X)η(Y)ξ-2g(X,Y)ξ

Subtracting (3.9) from (3.10) we get _ _ _ _ _ 2(Xγφ)X = a{2g(X, Y)ξ - η(Y)φX - η(X)φY} - β{η(X)φY + η(Y)φX}

-2η(X)η(Y)ξ - 2g(X, Y)ξ-XχφY + X_γφX - h(X, φY) + h(Y,φX)+φ[X,Y]

Hence Lemma is proved.

Lemma 3. Let M be a CR-submanifold of a nearly trans-hyperbolic Sasakian manifold _

M with a quarter symmetric semi metric connection, then

2(Xγφ)(Z) = A_φγZ - A_φzY - X_yφY + XφφZ - η(Y)Z + η(Z)Y - φ[Y,Z]

+a{2g(Y,Z) - η(Y)φZ - η(Z)φY} - β{η(Y)φZ + η(Z)φY}

-2η(Y')η(Z)ξ-2g(Y,Z)ξ

2(V _zφ)Y = a{2g(Y,Z)ξ - η(Y)φZ - η(Z)φY} - β{η(Y)φZ + η(Z)φY} -2η(Y)η(Z)ξ - 2g (Y,Z)ξ - A_φγZ + A_φzY + η (Y) Z - η (Z)Y + ^φY -^φZ + φ[Y,Z]

for any Y, Z e D ^l .

Proof. From Weingarten formula (2.13), we have

• (3.11)  V _zφY - V_γφZ = A_φγZ - Az_zY + ^φZ - Y₇φY - η(Y)Z + η(Z)Y

Also, we have

• (3.12)   ^v _zφY - ^vγφZ = (Vγφ)Z - (^zφ)Y + φ[Y,Z]

From (3.11) and (3.12), we get

• (3.13)  (Vγ φ-Z - (^zφ)Y = A_φγZ - A_φzY + ⅛φZ - ⅛φ - η(Y)Z + η(Z)Y -

Φ [ Y,Z]

Also for nearly trans-hyperbolic Sasakian manifold M with a quarter symmetric semi metric connection, we have _ _ _ _ _

• (3.14)  (Vγφ)Z + (Vzφ)Y = a{2g(Y,Z)ξ - η(YφφZ - η(Z)φY} - β{η(Y)φZ +

η(Z)φY} -2η(Y)η(Z)ξ-2g(Y,Z)ξ

Adding (3.13) and (3.14), we get

2(^γφ)(Z) = Aφ_γZ - AφzY - ⅛φY + ⅛φZ - φ[Y,Z] - η(Y)Z + η(Z)Y +a{2g(Y,Z)ξ - (YY)φZ - η(Z)φY} - β{η(Y)φZ + η(Z)φY} -2η(Y)η(Z)ξ-2g(Y,Z)ξ

Subtracting (3.13) from (3.14) we get

2(Vzφ)Y = a{2g(Y,Zξξ - η(Y)φZ - η(Z)φY} - β{η(Y)φZ + η(Z)φY} -2η(Y)η(Z)ξ -2g(Y,Z)ξ - AφγZ + AφzY + ⅛φY - ^φZ + η(Y)Z-η(Z)Y + φ[Y,Z]

This proves our assertions.

Lemma 4. Let M be a CR-submanifold of a nearly trans-hyperbolic Sasakian manifold _

M with a quarter symmetric semi metric connection, then

_

2(V_xφ)Y = a{2g(X, Y)ξ - η(Y)φX - η(X)φY} - β{η(Y)φX + η(X)φY} -2η(XMY)ξ - 2g(X, Y)ξ - AφγX + ⅛φY - η(X)Y + η(X)η(Y)ξ  -VγφX - h(Y, XX- - φ∖X, Y]

_ _ _ _ _

2(Vγφ)X = a{2g(X, Y)ξ - η(Y)φX - η(X)φY} - β{η(Y)φX + η(X)φY} -2η(X)η(Y)ξ - 2g(X, Y)ξ + AφγX - ⅛φY + η(X)Y -η(X)η(Y)ξ  +V_γφX + h(Y, φX) + φ[X, Y]

for any X e D and Y e D ¹.

Proof. By using Gauss equation and Weingarten equation for X e Dand Y e D¹ respectively we get

• (3.15)  V _xφY - ‰φX = -A_φγX + X1φY - η(X)Y + ηYXMY)ξ - X_γφX -

h( Y, φX)

Also, we have

• (3.16)   V _xφY - V _γφX = (^_xφ)Y - (V _γφ)X + φ[X , Y]

From (3.15) and (3.16), we get

• (3.17)   (Vχφ)Y - (Xγφ)X = -A_φγX + X1φY - ((X)Y + η(X)η(Y)ξ

-X_γφX - h(Y, φX) - φ[X , Y]

Also for nearly trans-hyperbolic Sasakian manifold M with a quarter symmetric semi metric connection, we have _ _ _ _ _

• (3.18)  (Xχφ )Y + (Xγφ )X = a{2g VY Y)ξ - (YYφφX - (VVΦY} - β{η(X)ΦY +

η(Y)φX} -2η(X)η(Y)ξ - 2g(X, Y)ξ

Adding (3.17) and (3.18), we get

_

2(Xχφ)Y = a{2g (J, Y)ξ - η(Y)XX - ηWΦY} - β{η(Y)φX + (VVΦY} -2η(XMY)ξ - 2g(X, Y)ξ - Aφ_γX + ⅛φY - (VVY + η(XMY)ξ -VγφX - h(Y, XX) - φ{X , Y]

Subtracting (3.9) from (3.10) we get

_

2(Xγφ)X = a{2g(X, Y)ξ - η(Y)φX - (VO YY} - β{η(Y)φX + (VOΦY} -2η(X)η(Y)ξ - 2g(X, Y)ξ + AφγX - XφφY + η(X)Y -η(X)η(Y)ξ +XγφX + h(Y, φX) + φ[X, Y]

Hence Lemma is proved.

• 4.    Parallel Distributions

Definition. The horizontal (respectly, vertical) distribution D (respectly, D¹) is said to be parallel [1] with respect to the connection on M if X_x Y e D (respectly, X_zW e D ¹) for any vector field ,       (respectly, ,        ).

Proposition 1. Let M be a ξ-vertical CR-submanifold of a nearly trans-hyperbolic Sasakian manifold M with a quarter symmetric semi metric connection. If the horizontal distribution is parallel, then

• (4.1)    h(X,φ Y)=h( Y, XX) for all ,          .

Proof. Using parallelism of horizontal distribution D , we have

• (4.2)    X_xφY e D, X_γφX e D for any X, Y e D.

Thus using the fact that X = Q Y = 0 for Y e D( (3.2) gives

• (4.3)  BhXX, YQ = g XX, YQQξ for any X, Y e D.

Also, since

• (4.4)  φhXX , Y) = BhXX , YQ + ChXX, YQ,

then

• (4.5)  φhXX, Y) = g(X, YQQξ + ChXX, Y) for any X, Y e D.

Next from (3.3), we have

• (4.6)    hXX, φYQ + h(Y, XX) = 2hh(X, Y) = 2φhXX, Y) - 2g(X, YQQξ,

for any X, Y e D. Putting X = φX e D in (4.6), we get

• (4.7)  h(XX, φYQ + h(Y, φ²XQ = 2φh(φX , YQ - 2g XφX , YQQξ

or

• (4.8)  h(φX,φYQ - h(Y, XQ = 2φh(φX,YQ - 2gXφX,YQQξ

Similarly, putting Y = φY e D in (4.6), we get

• (4.9)  h(φY, φXQ - hXX, YQ = 2φhXX, φYQ - 2g(X, φYQQξ.

Hence from (4.8) and (4.9), we have

• (4.10)    φhXX, φYQ - φh(Y, φXQ = g XX, φYQQξ - g φφX, YQQξ

Operating φ on both sides of (4.10) and using φξ = 0, we get

• (4.11)  h(X,φ YQ=hX Y, φXQ

for all ,          .

Now, for the distribution D ^j, we prove the following proposition.

Proposition 2. Let M be a ξ-vertical CR-submanifold of a nearly trans-hyperbolic Sasakian Manifold M with a quarter symmetric semi metric connection. If the distribution is parallel with respect to the connection on , then

• ⁽4.1²⁾   Aφ_γZ ⁺ Xφ_zY e D^j for any ,        .

Proof. Let ,Z e D^j , then using Gauss and Weingarten formula (2.10), we obtain

• (4.13)    -AφzY + VjφZ - ηXYQZ + (XYQηXZQξ - AφγZ + ^φY - ηXZQY + ηXZQηXYQξ

= φV_γZ + φhXY, ZQ + φ^ _zY + φhXZ,YQ + a{2gXY, ZQξ - ηXYQZZ -ηXZQφY}  -β{ηXYQφZ + ηXZQφY} - 2ηXYQηXZQξ - 2gXY,ZQξ

for any ,        . Taking inner product with      in (4.13), we get

• (4.14)   g(Aγ_yZ, X+g(Az_zY, X) = gXV_γZ, φXQ + gXV_z Y, φXQ

If the distribution D^j is parallel, then V _γZ e D ^j and V_zY e D ^j, for any Y, Z e D ^j.

So from (4.14) we get

• (4.15)   g(AφγZ,X) + g(A_φzY, X) = 0 or (4.13)   g(A_φγZ + AφzY,X) = 0

which is equivalent to

(4.15)   A_φγZ + A_φzY e Df for any Y, Z e D^r

and this completes the proof.

Definition : A CR-submanifold M of a nearly trans-hyperbolic Sasakian Manifold M with a quarter symmetric semi metric connection is said to be totally geodesic if h(X,Y ) = 0 for X e D and Y e D¹.

It follows immediately that a CR-submanifold is mixed totally geodesic if and only if A_nX e D for each X e D and N e T¹..

Let X e D and Y e φD¹ f For a mixed totally geodesic «-vertical CR-submanifold M of a nearly trans hyperbolic Sasakian Manifold M with a quarter symmetric semi metric connection then from (2.9), we have _

( X _xφ)N = 0

Since X_xφN = ( X_xφ)N + φ{X_xNs so that X_xφN = φ(X_xN).

Hence in view of (2.13), we get

^χxφN = -A _φnX + ^χfφN = -φA _nX + φ^χNN

As A _nX e D, A-AnX ^c D , so φ^fNN = 0 if and only if ^χ _xφNe D.

Thus we have the following proposition.

Proposition 3. Let M be a mixed totally geodesic «-vertical CR-submanifold of a nearly trans hyperbolic Sasakian Manifold M with a quarter symmetric semi metric connection. Then the normal section N e D^ffl is D parallel if and only if ^χ_xφN e D for all          .

• 5.    Conclusion

The notion of CR- submanifolds of a nearly trans-hyperbolic Sasakian manifold with a quarter symmetric semi metric connection investigated which shows that the existence of a parallel distribution relating to «-vertical CR-submanifolds of a nearly trans-hyperbolic sasakian manifold with a quarter symmetric semi metric connection. Further I have tried to find the condition under which the distributions required by CR-submanifolds of a nearly trans-hyperbolic Sasakian manifold with a quarter symmetric semi metric connection are parallel are obtained. -parallel normal section have been also studied.

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